 Hello and welcome to the session. In this session we will discuss proportion, consider four quantities P, Q, R and S. If we have P is to Q is equal to R is to S then we say that P, Q, R and S are in proportion and we write it as P is to Q as R is to S. This shows that the four quantities P, Q, R and S are in proportion. The terms P and S are called the extremes. The terms Q and R are called the means and the term S is the fourth proportional to P, Q and R. And when four quantities are in proportion then the product of extremes is equal to the product of the means. This means P into S that is this is the product of extremes is equal to the product of the means which is Q, R. So when the four quantities P, Q, R, S are in proportion so this would mean that P upon Q is equal to R upon S and from here we can conclude that P S is equal to Q R that is product of extremes is equal to the product of the means when the four quantities are in proportion. Next let's discuss continued proportion more quantities are in continued proportion when the first is to the second as the second is to the third if we are given that the quantities P, Q, R, S are in continued proportion so this means that P is to Q that is the first quantity is to the second as the second quantity that is Q is to the third quantity that is R. This is equal to the third quantity which is R is to the fourth quantity which is S and this is equal to so on so we can say that P upon Q is equal to Q upon R is equal to R upon S is equal to so on. So if we have the three quantities say P, Q, R are in continued proportion then P is to Q as Q is to R this means that P is to Q is equal to Q is to R or you can say that P upon Q is same as Q upon R and from here we get Q square is equal to P R. Now here this quantity Q is the mean proportional between P and R is the third proportional to P and Q and also as we have Q square equal to P R this means that the product of extremes which is P and R is equal to the square of the mean is Q so Q square is equal to P R. If suppose we are given that P is the mean proportional between 4 and 16 try to find out P so this means 4 is to P is equal to P is to 16 since this P is the mean proportional between 4 and 16 so we can write it like this so this means 4 upon P is equal to P upon 16 or you can say that P square equal to 16 into 4 that is 64 which gives us P equal to square root of 64 which is equal to 8. So hence we have got the value of P as 8. Let us now discuss some important results about proportion our first result is if Q is equal to R is to S and we also have P is equal to R then this gives us Q is equal to S. Let us try to prove this now now as P is to Q is equal to R is to S so this means that P upon Q is equal to R upon S and from where we have P into S is equal to Q into R now and R are equal so this means that S is equal to Q that is we now have Q is equal to S also if we were given Q equal to S then P would be equal to R this is our result 1. Now let us see the second result according to which we have if P is to Q is equal to R is to S then Q is to P is equal to S is to R. P is to Q is equal to R is to S so this means that P upon Q is equal to R upon S so this means that 1 divided by P upon Q is equal to 1 divided by R upon S. Now 1 divided by P upon Q would give us Q upon P and 1 divided by R upon S would give us S upon R and this means that Q is to P is equal to S is to R. So if we have P is to Q is same as R is to S then Q is to P is also same as S is to R. We can also say that if 4 quantities are in proportion then they are also proportionals when taken inversely and this result is called invert and do. Now we proceed with the next result according to which we have if P is to Q is equal to R is to S then P is to R is equal to Q is to S. Let us see how we get this result. P is to Q equal to R is to S means that P upon Q is equal to R upon S. Now we multiply both sides by Q upon R so we have P upon Q into Q upon R is equal to R upon S into Q upon R so this Q Q cancels this R R cancels and we now have P upon R is equal to Q upon S that is P is to R is equal to Q is to S. So from this result we can say that if 4 quantities be proportionals then they are also proportionals when taken alternately and this result is called alternate and do. Now we move on to the next result that is result 4 which says if P is to Q is equal to R is to S then P plus Q is to Q is equal to R plus S is to S. Now P upon Q is equal to R upon S as P is to Q is equal to R is to S adding 1 on both the sides we have P upon Q plus 1 is equal to R upon S plus 1 and so this means that P plus Q upon Q is equal to R plus S upon S or you can say that P plus Q is to Q is same as R plus S is to S. So from this result we can say that if 4 quantities are in proportion then first together with the second is to second as the third together with the fourth is to fourth and this result is called compolendo. Now we move on to the next result which is result 5 is to Q is equal to R is to S then P minus Q is to Q is equal to R minus S is to S. Let us see how we get this result P upon Q is equal to R upon S as P is to Q is equal to R is to S. Now subtracting 1 from both sides we have P upon Q minus 1 is equal to R upon S minus 1. So we have P minus Q upon Q is equal to R minus S upon S which means that P minus Q is to Q is equal to R minus S is to S that is if 4 quantities be proportional then the excess of the first over the second is to second as the excess of the third over the fourth is to fourth and this is called dividendo. So we have one more result which says if P is to Q is equal to R is to S then P is to P minus Q is equal to R is to R minus S as we have that P upon Q is equal to R upon S on applying invertendo we have Q upon P is equal to S upon R. Now subtracting 1 from both sides we have Q upon P minus 1 is equal to S upon R minus 1 which gives us Q minus P upon P is equal to S minus R upon R. Again on applying invertendo we get P upon Q minus P is equal to R upon S minus R. So this means that P is to P minus Q rather than place of Q minus P we can write P minus Q is equal to R is to R minus S in place of S minus R we can write R minus S. And this relation is convertendo we have another result which says if P is to Q is equal to R is to S then P plus Q upon P minus Q is equal to R plus S upon R minus S. Now P upon Q is equal to R upon S this is given to us we have P plus Q upon Q is equal to R plus S upon S this is by applying componento. Now we also have the P minus Q upon Q is equal to R minus S upon S this is when we apply dividendo to this relation. Now when we divide these two relations we get P plus Q upon P minus Q is equal to R plus S upon R minus S. This is a very useful relation and this says that if four quantities are in proportion then the sum of the first and second is to their difference as the sum of the third and fourth is to their difference. And this result is called convertendo and dividendo. So this completes the relation hope you have understood the concept of proportion and some important results on proportion.